Nuprl Lemma : eu-seg-length-extend
∀[e:EuclideanPlane]. ∀[s:ProperSegment]. ∀[t:Segment]. (|s + t| = |s| + |t| ∈ {p:Point| O_X_p} )
Proof
Definitions occuring in Statement :
eu-add-length: p + q,
eu-length: |s|,
eu-seg-extend: s + t,
eu-proper-segment: ProperSegment,
eu-segment: Segment,
euclidean-plane: EuclideanPlane,
eu-between-eq: a_b_c,
eu-X: X,
eu-O: O,
eu-point: Point,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
eu-proper-segment: ProperSegment,
eu-seg-proper: proper(s),
eu-segment: Segment,
eu-seg-extend: s + t,
eu-seg2: s.2,
eu-seg1: s.1,
pi1: fst(t),
pi2: snd(t),
eu-mk-seg: ab,
all: ∀x:A. B[x],
top: Top,
euclidean-plane: EuclideanPlane,
prop: ℙ,
implies: P ⇒ Q,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q)
Lemmas referenced :
eu_seg1_mk_seg_lemma,
eu_seg2_mk_seg_lemma,
eu-segment_wf,
eu-proper-segment_wf,
euclidean-plane_wf,
eu-extend-property,
not_wf,
equal_wf,
eu-point_wf,
eu-extend_wf,
eu-add-length-between,
eu-congruent-iff-length,
and_wf,
eu-between-eq_wf,
eu-O_wf,
eu-X_wf,
eu-add-length_wf,
eu-length_wf,
eu-mk-seg_wf,
eu-congruent_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
lemma_by_obid,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isectElimination,
hypothesisEquality,
axiomEquality,
because_Cache,
dependent_set_memberEquality,
lambdaFormation,
independent_isectElimination,
equalitySymmetry,
independent_pairFormation,
equalityTransitivity,
setEquality,
applyEquality,
lambdaEquality,
equalityEquality,
independent_functionElimination
Latex:
\mforall{}[e:EuclideanPlane]. \mforall{}[s:ProperSegment]. \mforall{}[t:Segment]. (|s + t| = |s| + |t|)
Date html generated:
2016_05_18-AM-06_38_41
Last ObjectModification:
2015_12_28-AM-09_23_51
Theory : euclidean!geometry
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